By [1]} we have that, on \((\Sigma _a^+,d_\delta ^+)\) , for \(\mu ^+\) -a.e. \(\mathbf {i}^+\in \Sigma _a^+\) ,
\(\dim _{loc}(\Phi _{\mathbf {I}}(\mu ^+),\Phi _{\mathbf {I}}(\mathbf {i}^+)):=\lim _{r\rightarrow \infty } \frac{\log \mu ^+\circ \Phi _{\mathbf {I}}^{-1}(B(\Phi _{\mathbf {I}}(\mathbf {i}^+),r))}{\log r}=\frac{h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^+,\mathbf {i}^+)}{-\log \delta },\) 
where \(h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^+,\mathbf {i}^+)\)  is the local projection entropy of \(\mu ^+\)  at \(\mathbf {i}^+\)  under \(\Phi _{\mathbf {I}}\)  (see [1]}). Denote by \(h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^+)=\int _{\Sigma _a} h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^+,\mathbf {i}^+) \mu ^+(\mathrm {d}\mathbf {i}^+)\) . If \(\mu ^+\)  is also ergodic then \(\mu \)  is exact-dimensional with dimension \(\frac{h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^+)}{-\log \delta }\) . By [1]} applying \(\eta \)  to \(\zeta \)  we have for \(\mu ^+\) -a.e. \(\mathbf {i}^+\) ,
\(h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^+,\mathbf {i}^+) =h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^\zeta _{\mathbf {i}^+}).\) 
This implies that
\(\overline{\dim }_H \Phi _{\mathbf {I}}(\mu ^+) & = \mathrm {ess sup}_{\mu ^+} \dim _{loc}(\Phi _{\mathbf {I}}(\mu ^+),\Phi _{\mathbf {I}}(\mathbf {i}^+)) \\& = \mathrm {ess sup}_{\mu ^+} \frac{h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^+,\mathbf {i}^+)}{-\log \delta } \\& = \mathrm {ess sup}_{\mu ^+} \frac{h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^\zeta _{\mathbf {i}^+})}{-\log \delta } \\& = \mathrm {ess sup}_{\mu ^+} \dim \Phi _{\mathbf {I}}(\mu ^\zeta _{\mathbf {i}^+})\) 

and
\(\underline{\dim }_H \Phi _{\mathbf {I}}(\mu ^+) & = \mathrm {ess inf}_{\mu ^+} \dim _{loc}(\Phi _{\mathbf {I}}(\mu ^+),\Phi _{\mathbf {I}}(\mathbf {i}^+)) \\& = \mathrm {ess inf}_{\mu ^+} \frac{h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^+,\mathbf {i}^+)}{-\log \delta } \\& = \mathrm {ess inf}_{\mu ^+} \frac{h_{\Phi _{\mathbf {I}}}(\sigma ,\mu ^\zeta _{\mathbf {i}^+})}{-\log \delta } \\& = \mathrm {ess inf}_{\mu ^+} \dim \Phi _{\mathbf {I}}(\mu ^\zeta _{\mathbf {i}^+})\) 
